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Hey there, Guest!

So, let's get to it:

\(\frac{(6m-5)}{(2m^2-m-6)}+\frac{(2-3m)}{(m^2-4)}\)

\(\frac{2m^2-8m+8}{2m^4-m^3-14m^2+4m+24}\)

\(\frac{2(m-2)(m-2)}{(m+2)(m-2)(m-2)(2m+3)}\)

\(=\frac{2}{2m^2+7m+6}\)

And there's our answer.

Hope this helped! :)

( ﾟдﾟ)つ Bye

Here you go.

HELP

Just for you.

Here is a process to solve this problem:

Hope this helps. Also, if you ever need more help on Algebra problems like this, I recommend using photomath on your phone, it gives steps in detail on how to solve these types of problems.

Here is what you need to do for the first one:

Using calculus, the minimum is sqrt(2) - 2.

Same thing as last time. Pull out 15 from both - a1 = 1, a3 = 9.

Ask: What can I multiply 1 by twice to get 9?

\(1 * x * x = 9\)

\(9 = x^2, x = 3\)

Okay so if a geometric equation is of form:

\(y=ab^x\)

Then a = 15 (what we pulled out to make the problem easier to solve) and b = 3

\(y = 15(3^x)\)

There would be 40 paths.

From B to C would be 4C1 or 4C3 which would give you 4. (whichever combination you choose will give you the same answer)

From A to C would be 5C2 or 5C3 which would give you 10.

Now multiply the outcomes:

4*10=40

The first step is to find the slope:

slope formula: y₂-y₁/x₂-x₁

If you plug the values in the slope would be: -28/3

This is how your equation should be looking so far:

y=-28/3x+b

Now, plug one of the pairs into the equation and solve for b.

I'll use (-5,11)

11=140/3+b

b=-107/3

Your final equation would be y=-28/3x-107/3

Your expression looks like a geometric sequence of powers of 2. This is helpful because if we want to express something as a power of two, we need to first get all the base numbers to be 2.

We'll use our first power rule, that \({(2^a)}^b = 2^{a+b}\)

For example, \(8^2 = {2^3}^2 = 2^{3*2} = 2^6\)

We get \(2^2 * 2^4 * 2^ 6 * ... * 2^{20}\)

Now we get to use another power rule, that \(2^a * 2^b = 2^{a+b}\)

\(2^{2+4+6+...+20} = 2^{110}\)

As a quick tip, you can do that addition quickly by remembering that in a linearly increasing sequence, the sum = (first + last)/2 * # of terms

By complementary counting, there are 408 possible ways.

Let's call 1970 our zero point. We use that to determine our zero point.

If a linear equation is \(y = mx + b\)

Then for us, \(b=94,000\)

(Note that I made 1970 our zero point for convenience)

Now, what's the change in population size our zero point? Well it's gone from 94,000 to 161,000, and it took 22 years. So the difference between those two numbers is our rise and 22 years is our run.

\(161,000 - 94,000 \over 1992 - 1970\)

\({66,000 \over 22} = 3,000\)

Rise over run is slope, or m. So \(m = 3,000\).

So our equation is \(y = 3,000x + 94,000\), where x is the years since 1970.

2035 is 65 years after 1970:

\(y = 3,000 * 65 + 94,000 = 195,000 + 94,000 = 289,000\)

There would be a total of 126 paths.

You can either do 9C4 or 9C5, both would get you to 126.

First thing I see is that we can take 12 out of both of them, that's helpful.

a2 is 1, a4 is 4.

2^0 is 1, and 2^2 is 4, that's a helpful fact... but our subscripts were 2 and 4, not 0 and 2. We could shift them by using 2^(x - 2). Thus:

\(12 * 2^{x-2}\)

I suppose 2^x * 2^-2 is basically 2^x / 4, so we could simplify further to:

\(3 * 2^{x}\)

CPhill how do you have the time to help so many people on here?! Honestly, it's amazing!

Oops, my bad. 

My brain stopped working for a moment.

Thank you for correcting me. :))

=^._.^=

It seems pretty appropriate. It's harder for the penny to win but it also gets an additional flip. If you kindly refer back to the question, the asker specifies 3 flips for penny and 2 flips for nickels.

It seems a bit weird that nickel has equal heads as pennies. 

I think you forgot the scenerio where nickel has 3 heads. 

=^._.^=

Well, we need to determine how many outcomes of flipping would result in a penny "win" vs a penny "tie or loss".

For a penny, there's 2^3 possibilities (number of options to the power of number of flips), and for nickels it'd be 2^2. The chance of getting each result is then "x in 4" or "x in 8".

So add the results of the penny runs that beat the each nickel run together:

If Nickels get zero, Pennies win 7 in 8 runs

If Nickels get one, Pennies win 4 in 8 runs

if Nickels get two, pennies win 1 in 8 runs

Multiply each of these by its weight (i.e. how likely the scenario is to occur):

7 in 8 * 1 in 4 ||\(7/8 * 1/4 = 7/32\)

4 in 8 * 2 in 4 || \(4/8 * 2/4 = 1/4 or 8/32\)

1 in 8 * 1 in 4 || \(1/8 * 1/4 = 1/32\)

So I see 16 in 32 or simply 1 in 2.

Half the time, Penny wins :)

Well, FV (future value) & PV (present value) are related as follows:

\(FV = PV (1 + r)^t\)

And interest is really just the future value of an investment minus the present value. r is your rate, t is the time that passed, which is conveniently 1 in this example.

Thus your increase due to interest is \(1+r , or, 1.005\) (since we have to divde percentages by 100).

\(10,000,000 * 1.005 =10,050,000\)

\(10,050,000 - 10,000,000 = 50,000\)

^ Interest paid by the bank

The second half could be done the very same way. Or, we can notice that since only 1 year passed and the interest is calculated annually, we can skip some steps and just multiply the interest rate by the present value:

\(10,000,000 \times .0465 = 465,000\)

As is typically, the bank is pulling one over on the US gov't. That deal cost $415,000!

If you're ever doing "net" costs like this, it might be helpful to know you can skip one more step and just subtract the two rates from each other before multiplying by your present value (or principle):

\(10m \times .0465 - 10m \times .005 = 10m (.0465 - .005)\)

Slope  =  ( 5 - - 11)  /( -4 - 1)   =  16 / -5   =  -16/5

Equation

y  =  (-16/5)(x -1)  -  11                multiply through by  5

5y  =  -16( x - 1)  -  55

5y  = -16x  + 16  -  55

5y  =  - 16x  -  39          divide through  by  -39      and  rearange  as

(-5/39)y - ( 16/39)x  =  1

x / ( -39/16)  +  y / ( -39/5 )  =   1

a  =  -39/16

Slope intercept form of a line    y = mx + b     where m = slope and b = y axis intercept

y =  -3/7 x  + 18         the x axis intercept occurs when y = 0

0 = - 3/7x + 18

x = 18 * 7/3 = 42                42, 0   is the x axis intercept

6|4−3d|−2|d+5| > 9

We have two equations to solve

abs  ( 24 - 18d)  - abs ( 2d + 10)  > 9

We can write                                 

24 - 18d -  ( 2d + 10) > 9                   18d - 24  - ( 2d + 10) > 9

14  - 20d  >  9                                   16d   -  34 >  9

-20d  >  9 -  14                                  16d  >  43

-20 d >   -5                                           d >   43/16

d <   -5/-20

d < 1/4

So....the solution set is

(-inf , 1/4)  U  (43/16, inf)

Find the slope     rise/run = - 2

y axis intercept = -3

slope intercept form of the line becomes    y = -2x-3

re-arrange to the requested form

2x + y = -3

If  10  dropped  and the  number  who  took neither increased from 27  to 31.....then 4 who dropped  must have only been taking  Chinese......then  the  other  6  must  have  been  taking Spanish and Chinese

So....the  number  who  take only Spanish  must increase by 6